Singularities in General Relativity

广义相对论中的奇点

• 批准号: 1709458
• 负责人: Jonathan Luk
• 金额: $ 12万
• 依托单位: Stanford University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-07-01 至 2020-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1709458&HistoricalAwards=false
• 关键词: Singularities; General; Relativity

This research project studies spacetime singularities in the solutions to the Einstein vacuum equations, a system of nonlinear partial differential equations that describes gravity in locations devoid of matter. Solutions to this system of equations can develop singularities from regular initial data; some such singularities are inside a black hole event horizon. A fundamental mathematical question concerns understanding how such singularities arise from generic initial data. In particular, this project explores questions that include: What does the boundary of the interior of black holes look like? Are they always singular? Are there singularities that are not in the black hole regions? These are questions that the strong and weak cosmic censorship conjectures in general relativity seek to address -- study of those conjectures is the central motivation for this project.A particular focus of the project is to understand the singular boundary of the interior of black holes and also to understand sharp criteria under which trapped surfaces (structures closely related to black holes) may form. Recent work of the investigator and collaborators provides a resolution of the strong cosmic censorship conjecture for the Einstein-Maxwell (real) scalar field in spherical symmetry, showing that solutions arising from generic data have singular black hole interiors, in the sense that the spacetime metric cannot be extended in a suitably regular manner. This project aims to extend these results to settings without symmetry assumptions, in particular to give a complete description of the boundary of black holes that arise from perturbations of the explicit Kerr solutions. In a different direction, this project also aims to establish criteria that guarantee that trapped surfaces form, as a step towards understanding the weak cosmic censorship conjecture in general relativity.

这个研究项目研究爱因斯坦真空方程解中的时空奇点，爱因斯坦真空方程是一个非线性偏微分方程系统，描述了没有物质的地方的重力。该方程组的解可以从正则初始数据发展为奇点；一些这样的奇点位于黑洞事件视界内。一个基本的数学问题是如何理解这些奇点是如何从一般的初始数据中产生的。特别是，这个项目探索的问题包括：黑洞内部的边界是什么样子的？它们总是单数吗？有没有不在黑洞区域的奇点？这些都是广义相对论中强宇宙审查猜想和弱宇宙审查猜想试图解决的问题——研究这些猜想是这个项目的核心动机。该项目的一个特别重点是了解黑洞内部的单一边界，以及了解可能形成的捕获表面（与黑洞密切相关的结构）的尖锐标准。研究者和合作者最近的工作为球面对称的爱因斯坦-麦克斯韦（实）标量场提供了强宇宙审查猜想的解决方案，表明由一般数据产生的解具有奇异的黑洞内部，从某种意义上说，时空度规不能以适当的规则方式扩展。该项目旨在将这些结果扩展到没有对称假设的情况下，特别是给出由显式克尔解的扰动引起的黑洞边界的完整描述。在一个不同的方向上，这个项目还旨在建立保证被困表面形成的标准，作为理解广义相对论中弱宇宙审查猜想的一步。

项目成果

The interior of dynamical extremal black holes in spherical symmetry

球对称动力学极值黑洞的内部

• DOI: 10.2140/paa.2019.1.263
• 发表时间: 2019
• 期刊: Pure and Applied Analysis
• 作者: Gajic, Dejan;Luk, Jonathan
• 通讯作者: Luk, Jonathan

Stability of Vacuum for the Landau Equation with Moderately Soft Potentials

• DOI: 10.1007/s40818-019-0067-2
• 发表时间: 2018-07
• 期刊: Annals of PDE
• 影响因子: 2.8
• 作者: J. Luk